# Properties

 Label 8T31 Order $$64$$ n $$8$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $(((C_4 \times C_2): C_2):C_2):C_2$

# Related objects

## Group action invariants

 Degree $n$ : $8$ Transitive number $t$ : $31$ Group : $(((C_4 \times C_2): C_2):C_2):C_2$ CHM label : $[2^{4}]E(4)$ Parity: $-1$ Primitive: No Generators: (1,3)(2,8)(4,6)(5,7), (1,8)(2,3)(4,5)(6,7), (4,8) $|\Aut(F/K)|$: $2$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $V_4$ x 7
8:  $D_{4}$ x 6, $C_2^3$
16:  $D_4\times C_2$ x 3
32:  $V_4 \wr C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $V_4$

## Low degree siblings

8T29 x 6, 8T31, 16T127, 16T128 x 3, 16T129 x 3, 16T147, 16T149 x 6, 16T150 x 3, 32T136 x 3, 32T137 x 2, 32T163 x 3

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 1, 1, 1, 1, 1, 1$ $4$ $2$ $(4,8)$ $2, 2, 1, 1, 1, 1$ $2$ $2$ $(3,7)(4,8)$ $2, 2, 1, 1, 1, 1$ $2$ $2$ $(2,6)(4,8)$ $2, 2, 1, 1, 1, 1$ $2$ $2$ $(2,6)(3,7)$ $2, 2, 2, 1, 1$ $4$ $2$ $(2,6)(3,7)(4,8)$ $2, 2, 2, 2$ $4$ $2$ $(1,2)(3,4)(5,6)(7,8)$ $4, 2, 2$ $8$ $4$ $(1,2)(3,4,7,8)(5,6)$ $4, 4$ $4$ $4$ $(1,2,5,6)(3,4,7,8)$ $2, 2, 2, 2$ $4$ $2$ $(1,3)(2,4)(5,7)(6,8)$ $4, 2, 2$ $8$ $4$ $(1,3)(2,4,6,8)(5,7)$ $4, 4$ $4$ $4$ $(1,3,5,7)(2,4,6,8)$ $2, 2, 2, 2$ $4$ $2$ $(1,4)(2,3)(5,8)(6,7)$ $4, 2, 2$ $8$ $4$ $(1,4,5,8)(2,3)(6,7)$ $4, 4$ $4$ $4$ $(1,4,5,8)(2,3,6,7)$ $2, 2, 2, 2$ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$

## Group invariants

 Order: $64=2^{6}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [64, 138]
 Character table:  2 6 4 5 5 5 4 4 3 4 4 3 4 4 3 4 6 1a 2a 2b 2c 2d 2e 2f 4a 4b 2g 4c 4d 2h 4e 4f 2i 2P 1a 1a 1a 1a 1a 1a 1a 2b 2i 1a 2c 2i 1a 2d 2i 1a 3P 1a 2a 2b 2c 2d 2e 2f 4a 4b 2g 4c 4d 2h 4e 4f 2i X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 1 X.3 1 -1 1 1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 X.4 1 -1 1 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 X.5 1 -1 1 1 1 -1 1 -1 1 1 -1 1 1 -1 1 1 X.6 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 X.7 1 1 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 X.8 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 X.9 2 . 2 -2 -2 . -2 . 2 . . . . . . 2 X.10 2 . 2 -2 -2 . 2 . -2 . . . . . . 2 X.11 2 . -2 -2 2 . . . . . . . -2 . 2 2 X.12 2 . -2 -2 2 . . . . . . . 2 . -2 2 X.13 2 . -2 2 -2 . . . . -2 . 2 . . . 2 X.14 2 . -2 2 -2 . . . . 2 . -2 . . . 2 X.15 4 -2 . . . 2 . . . . . . . . . -4 X.16 4 2 . . . -2 . . . . . . . . . -4